Optimal. Leaf size=111 \[ \frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 \left (c d^2-a e^2\right ) (d+e x)^2}+\frac {4 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 \left (c d^2-a e^2\right )^2 (d+e x)} \]
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Rubi [A]
time = 0.04, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {672, 664}
\begin {gather*} \frac {4 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 (d+e x) \left (c d^2-a e^2\right )^2}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 (d+e x)^2 \left (c d^2-a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 664
Rule 672
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 \left (c d^2-a e^2\right ) (d+e x)^2}+\frac {(2 c d) \int \frac {1}{(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 \left (c d^2-a e^2\right )}\\ &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 \left (c d^2-a e^2\right ) (d+e x)^2}+\frac {4 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 \left (c d^2-a e^2\right )^2 (d+e x)}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 61, normalized size = 0.55 \begin {gather*} \frac {2 \sqrt {(a e+c d x) (d+e x)} \left (-a e^2+c d (3 d+2 e x)\right )}{3 \left (c d^2-a e^2\right )^2 (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.71, size = 131, normalized size = 1.18
method | result | size |
trager | \(-\frac {2 \left (-2 c d e x +e^{2} a -3 c \,d^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{3 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (e x +d \right )^{2}}\) | \(81\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-2 c d e x +e^{2} a -3 c \,d^{2}\right )}{3 \left (e x +d \right ) \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}\) | \(89\) |
default | \(\frac {-\frac {2 \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{3 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {4 c d e \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{3 \left (e^{2} a -c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )}}{e^{2}}\) | \(131\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.29, size = 141, normalized size = 1.27 \begin {gather*} \frac {2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + 3 \, c d^{2} - a e^{2}\right )}}{3 \, {\left (2 \, c^{2} d^{5} x e + c^{2} d^{6} - 4 \, a c d^{3} x e^{3} + a^{2} x^{2} e^{6} + 2 \, a^{2} d x e^{5} - {\left (2 \, a c d^{2} x^{2} - a^{2} d^{2}\right )} e^{4} + {\left (c^{2} d^{4} x^{2} - 2 \, a c d^{4}\right )} e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.25, size = 177, normalized size = 1.59 \begin {gather*} -\frac {4 \, \sqrt {c d} c d e^{\frac {1}{2}} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{3 \, {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}} + \frac {2 \, {\left (3 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} c d e - {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}}\right )}}{3 \, {\left (c d^{2} e^{2} - a e^{4}\right )} {\left (c d^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - a e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.72, size = 69, normalized size = 0.62 \begin {gather*} \frac {2\,\left (3\,c\,d^2+2\,c\,x\,d\,e-a\,e^2\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{3\,{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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